Initial orbit determination using angular velocity and angular acceleration measurements

ABSTRACT

The current disclosure provides systems and methods of directly measuring angular velocity and angular acceleration of space objects and using the measured angular velocity and angular acceleration as inputs into new and unique algorithms for initial orbit determination. Sensors measuring locations and times of light events may be used to generate a virtual rate track image for identification of space objects. Right ascension and declination of the space object events versus time may be fit to polynomials or splines to determine associated angle, angular rate, and angular acceleration of the space objects. New and unique initial orbit algorithms may then be applied to estimate orbital elements of the space objects.

RELATED APPLICATIONS

This non-provisional patent application claims priority benefit, withregard to all common subject matter, of earlier-filed U.S. ProvisionalPatent Application No. 63/389,538 filed Jul. 15, 2022, and entitledINITIAL ORBIT DETERMINATION USING ANGULAR VELOCITY AND ANGULARACCELERATION MEASUREMENTS. The identified earlier-filed provisionalpatent application is hereby incorporated by reference in its entiretyinto the present application.

BACKGROUND 1. Field

Embodiments of the disclosure relate to initial orbit determination(IOD). More specifically, embodiments of the disclosure relate to IODusing measured angular rate and measured angular acceleration as inputs.

2. Related Art

Typically, standard orbits require six parameters to be described:eccentricity, semimajor axis, inclination, longitude of the ascendingnode, argument of periapsis, and true anomaly. These parameters may bedetermined by taking measurements of orbiting bodies over time andutilizing orbit determination algorithms to estimate the orbits.Standard methods involve a plurality of sensors for measuring positionat various times, calculating physical model trajectories, projectingpotential orbits, and converging on an estimated orbit. These typicalmethods generally take a relatively long time (e.g., minutes or hours)to obtain an initial estimate of the orbit of the object.

Traditionally, various IOD approaches have been performed withrelatively similar results. One class of traditional IOD techniques isto collect a plurality of observations over relatively long periods(e.g., the method of Gooding). Trial orbital path generations may beintegrated over time to compare to observed data. The trial orbits areiterated upon until the mid-point observation matches the measuredazimuth and elevation for the time of collection for the middleobservation. When the observation and trial orbit align, an accurateinitial orbit is determined. However, any change in orbital parametersof the satellite during the collection period will result in inaccurateresults.

A second exemplary IOD calculation method typically called the Laplacemethod uses three discrete observations to numerically estimate angularvelocity and angular acceleration at the mid-point observation. Aproblem associated with this method is that the angular velocity andangular acceleration of both the observer and target can changedramatically between observations. If the time between the successiveobservations is too short, errors in angular measurement will lead toinaccurate numerical derivatives of the angular velocity and angularacceleration. If the time between the successive observations is toolong, the angular velocity and angular acceleration of the observer andtarget object may change too much for an accurate estimate to be madefor the middle observation point. The objects motion may also have asignificant third derivative (change in angular acceleration), whichmakes the numerical estimate inaccurate.

Currently, more and more satellites and more and more space junk arebeing placed into orbit. This creates clutter for satellites. Thisclutter must be tracked to avoid potential impacts. When satellites aremoved by commanded or uncommanded forces new orbit updates must beperformed. The longer it takes to determine the new orbit, the morelikely it is that an impact will take place. Therefore, it is necessaryto develop new techniques for IOD that are quick and accurate.

There are currently plans to send more satellites into lunar orbits.These lunar/earth orbits can be unconventional and can be changed bycommanded and uncommanded forces (e.g., relatively strong solarradiation pressure). It is necessary to accurately track these orbitchanges to take corrective measures.

What is needed are systems and methods of quickly and accuratelyperforming IOD.

SUMMARY

Embodiments of the invention solve the above-mentioned problems byproviding systems and methods of directly measuring angular velocity andangular acceleration and using the measured angular velocity and angularacceleration as inputs into new and unique algorithms for IOD.

A first embodiment is directed to a method of determining an orbit of anobject. The method comprises obtaining, by a sensor, data indicative ofa plurality of objects in orbits, wherein the data is indicative ofangular velocity and angular acceleration of the object of the pluralityof objects, identifying the object from the data, measuring the angularvelocity and the angular acceleration from the data, determining aninitial orbit of the object using the angular velocity and the angularacceleration, and estimating orbital elements from the initial orbit ofthe object.

A second embodiment is directed to one or more non-transitorycomputer-readable media storing computer-executable instructions that,when executed by at least one processor, performs a method ofdetermining an orbit of an object. The method comprises obtaining, by asensor, data indicative of a plurality of objects in orbits, wherein thedata is indicative of angular velocity and angular acceleration of theobject of the plurality of objects, identifying the object from the databy generating a virtual rate track image comprising virtual rate tracksof each object of the plurality of objects in the virtual rate trackimage, mapping the virtual rate tracks to locations in a field of view,identifying the plurality of objects based on the locations in the fieldof view, and identifying the object from the plurality of objects basedon a comparison of the virtual rate tracks, and determining the angularvelocity and the angular acceleration of the object from the virtualrate tracks, determining an initial orbit of the object using theangular velocity and the angular acceleration, and estimating orbitalelements from the initial orbit of the object.

A third embodiment is directed to one or more non-transitorycomputer-readable media storing computer-executable instructions that,when executed by at least one processor, performs a method ofdetermining an orbit of an object. The method comprises obtaining, by asensor, data indicative of a plurality of objects in orbits, wherein thedata is indicative of angular velocity and angular acceleration of theobject of the plurality of objects, identifying the object from thedata, measuring the angular velocity and the angular acceleration fromthe data, determining an initial orbit of the object by fitting an orbitephemeris to the data, and determining the orbit of the object using theinitial orbit as a starting seed solution for an orbit determinationalgorithm.

This summary is provided to introduce a selection of concepts in asimplified form that are further described below in the detaileddescription. This summary is not intended to identify key features oressential features of the claimed subject matter, nor is it intended tobe used to limit the scope of the claimed subject matter. Other aspectsand advantages of the invention will be apparent from the followingdetailed description of the embodiments and the accompanying drawingfigures.

BRIEF DESCRIPTION OF THE DRAWING FIGURES

Embodiments of the invention are described in detail below withreference to the attached drawing figures, wherein:

FIG. 1 depicts exemplary prior art methods of IOD;

FIGS. 2A and 2B depict exemplary embodiments of determining an initialstate vector using angular velocity and angular acceleration;

FIG. 3 depicts exemplary sensors for measuring angular velocity andangular acceleration;

FIG. 4 depicts an embodiment of a photon counting imager vs framingsensor imaging;

FIG. 5 depicts an exemplary satellite detection by an event-basedsensing system;

FIG. 6 depicts an embodiment of a system comprising a telescope and aRonchi filter for measuring angular velocity and angular acceleration;

FIG. 7 depicts an embodiment of utilizing a color sensor and filters forindependent measurements of two-dimensional motion and brightness;

FIG. 8 depicts an exemplary process of IOD;

FIG. 9 depicts an exemplary process of calibration of a sensor for IOD;and

FIG. 10 depicts a hardware system for embodiments of the disclosure.

The drawing figures do not limit the invention to the specificembodiments disclosed and described herein. The drawings are notnecessarily to scale, emphasis instead being placed upon clearlyillustrating the principles of the invention.

DETAILED DESCRIPTION

The following detailed description references the accompanying drawingsthat illustrate specific embodiments in which the invention can bepracticed. The embodiments are intended to describe aspects of theinvention in sufficient detail to enable those skilled in the art topractice the invention. Other embodiments can be utilized, and changescan be made without departing from the scope of the invention. Thefollowing detailed description is, therefore, not to be taken in alimiting sense. The scope of the invention is defined only by theappended claims, along with the full scope of equivalents to which suchclaims are entitled.

In this description, references to “one embodiment,” “an embodiment,” or“embodiments” mean that the feature or features being referred to areincluded in at least one embodiment of the technology. Separatereferences to “one embodiment,” “an embodiment,” or “embodiments” inthis description do not necessarily refer to the same embodiment and arealso not mutually exclusive unless so stated and/or except as will bereadily apparent to those skilled in the art from the description. Forexample, a feature, structure, act, etc. described in one embodiment mayalso be included in other embodiments but is not necessarily included.Thus, the technology can include a variety of combinations and/orintegrations of the embodiments described herein.

Generally, embodiments of the current disclosure are directed to systemsand methods for quickly and accurately performing initial orbitdetermination (IOD). In some embodiments, sensors are used for measuringangular velocity and angular acceleration and the angular velocity andthe angular acceleration are used as inputs into new and unique orbitdetermination algorithms. Optical sensors may be used to take a singleoptical measurement of the sky comprising the object. Utilizing theoptical sensors along with new techniques for object detection and orbitdetermination allows for the transformation of object track datacomprising time-tagged pixel illumination and/or photon position datainto angular position, angular velocity, and angular acceleration datathat may be used to generate an initial object state vector. The systemsand methods described herein may quickly and accurately provide aninitial orbit solution as a seed for estimating the orbit of new andmaneuvered space objects.

Generally, orbits are described in terms of six parameters (i.e.,eccentricity, semimajor axis, inclination, longitude of the ascendingnode, argument of periapsis, and true anomaly). Typically, two anglesare measured. In order to describe an orbit these six parameters must beknown. Therefore, the object must be tracked to obtain enoughinformation to calculate these parameters. Tracking to obtain enoughinformation to accurately estimate the orbit of the space objects cantake minutes or hours. The only requirements are that the motion of theobserver and that the gravitational potential along the line of sight beknown.

Typically, angular rate and angular acceleration are calculated from thefixed angle measurements. Therefore, the angular velocity and angularacceleration are coupled to the position measurements. Consequently,errors in the position measurements are extended to the angular velocityand angular acceleration calculations. Furthermore, these parameters arecoupled in the covariance matrix of current orbit determination methodscausing error minimization and noise reduction to be furthercomplicated. Utilizing the sensors described herein, the angularvelocity and angular acceleration can be directly measured decouplingangular velocity and angular acceleration from the position measurementsof the space objects. Using the angular velocity and the angularacceleration derived directly from measurements as inputs into initialorbit determination algorithms allows the orbit of the space object tobe more quickly and accurately determined than typical standard methods.

In some embodiments, the space objects may be any object in orbit aboutany other object (e.g., planets, stars, satellites, missiles, spacejunk, and unknown objects). Furthermore, the measurement may be takenfrom the observer, which may be positioned on the earth or any otherobject in orbit. In this way, the orbit of any object may be determinedfrom any other object. Accordingly, it may be determined if any two ormore orbits intersect. Furthermore, it may be quickly and accuratelydetermined where an orbit may intersect with a location on earth or anyother planet or satellite. Any circular, elliptical, parabolic, orhyperbolic orbit may be quickly and accurately be estimated. Similarly,any irregular orbit or unknown trajectory about any celestial bodyincluding the moon or comprising the two-body problem, three-bodyproblem, or more, may be estimated. Furthermore, any standard altitudeorbit (i.e., Low Earth Orbit (LEO), Medium Earth Orbit (MEO), HighElliptical Orbit (HEO), Geosynchronous Earth Orbit (GEO)) may beestimated using the systems and methods described herein.

In some embodiments, unconventional orbits may be estimated using thesystems and methods described herein. There are currently new efforts toexpand orbits out to the moon and to other planets. This results inunconventional orbits that do not adhere to the two-body convention.This raises new uncertainty for space objects. The methods and systemsdescribed herein can also estimate these unconventional orbits (e.g.,lunar orbits, L2 region, earth-moon orbits, orbit transitions, nearrectilinear halo).

Generally, orbits may be determined using orbit estimation algorithmsfrom measured data. Typically, state vectors are determined usingparameters illustrated in FIG. 1 . For example, traditional methods ofIOD are shown in FIG. 1 . FIG. 1 depicts an exemplary estimation ofrange (ρ) and range rate ({dot over (ρ)}) known as the Laplace method102. As shown, angular velocity ({dot over (u)}₂) and angularacceleration (ü₂) cannot be known and must be numerically estimated fromthe observed position vectors (û₁, û₂, and û₃) position information attimes (T₁, T₂, and T₃). If the observations are too close, errors inobservation angles produce too much error in the numerical angularderivative motion estimates. If the observations are too far apart, thetrue angular velocity and the true angular acceleration changes too muchbetween the three measurements for the numerical derivative estimates tobe accurate at the mid-time observation.

The Laplace method 102 has its drawbacks. It is currently the leastaccepted method of IOD. The angular derivatives, angular velocity ({dotover (u)}₂) and angular acceleration (ü₂), must be numerically estimatedfrom the observed position vectors (û₁, û₂, and û₃). Therefore, theangular velocity and the angular acceleration must be derived frominterpolations of fixed angle measurements. Because the angular velocityand angular acceleration values must be derived from fixed anglemeasurements, the angular velocity and angular acceleration will alwaysbe coupled to the fixed angle measurements. This couples any error inmeasurements, enhances error in measurements, and makes minimizing noisedifficult.

The more accepted method is the method of Gooding 104. The method ofGooding 104 comprises collecting a plurality of observations overrelatively long periods. Trial orbital path generations may beintegrated over time to compare to observed data. The trial orbits areiterated upon until the mid-point observation matches the measuredazimuth and elevation for the time of collection for the middleobservation. When the observation and trial orbit align, an accurateinitial orbit is determined. However, any change in orbital parametersof the satellite during the collection period will result in inaccurateresults.

The application of new sensors can allow angular velocity and angularacceleration to be measured rather than calculated. Measuring theangular velocity and angular acceleration decouples the angular velocityand the angular acceleration from the position measurements. Theseindependent measurements add in two terms that previously had to becalculated through successive measurements of fixed angles. Becauseangle, angular velocity, and angular acceleration can be measuredsimultaneously, the time to converge on an estimated initial statevector and, thus, an estimated orbit is reduced relative to priormethods. FIGS. 2A and 2B depict exemplary embodiments of using fittedangle, angular rate, and angular acceleration values to find an initialstate vector.

FIG. 2A and FIG. 2B depict embodiments of the free-body diagram andcalculations required in typical methods to determine range and rangerate from measured angular velocity and angular acceleration from anearth-based observer (FIG. 2A) and a space-based observer (FIG. 2B).Here, in embodiments described herein, angular velocity and angularacceleration do not need to be numerically estimated. This reducescomputation leaving a single non-linear equation for calculating rangeand range rate as shown below and derived in FIG. 2B.

${{{\rho\left( {\hat{u} \times \overset{.}{u}} \right)} \cdot \overset{¨}{u}} - {\left( {\hat{u} \times \overset{.}{u}} \right) \cdot {\overset{\rightarrow}{A}(\rho)}}} = 0$$\overset{˙}{\rho} = \left( \frac{{\overset{.}{u} \cdot {\overset{\rightarrow}{A}(\rho)}} - {\rho\left( {\overset{.}{u} \cdot \overset{¨}{u}} \right)}}{2\left( {\overset{.}{u} \cdot \overset{¨}{u}} \right)} \right)$

Therefore, the initial target state vector for the orbiting object canbe determined. Similarly, any method of IOD may be adapted to includeinputs of the measured angular velocity and angular acceleration andparameters determined therefrom (e.g., range and range rate). Usingthese measured values decouples angular acceleration and angularvelocity from the position measurements in the initial state vector. Theinitial state vector may then be used as input into orbit determinationalgorithms such as a Kalman filter (KF), extended Kalman filter (EKF),Batch filters and the like, as described in embodiments below. Using themeasured angular rates in calculation of the initial state vector as astarting point for the exemplary EKF amplifies the EKF to converge on asolution much more quickly; in some embodiments, an order of magnitudefaster than traditional methods.

FIG. 3 depicts an exemplary collection of sensors 300 that may be usedin embodiments described herein. Ideally, a sensor that detects time ofarrival and spatial location on the focal plane as well as thewavelength and/or polarization for each light event would provideinformation for greatly increasing the accuracy and reliability formeasurements such that quick and accurate IOD would be possible.Currently, there is a tradeoff between measuring spatially andtemporally for framing sensors. When high speed imaging is used, thelocation of the object includes error because a relatively low number ofphotons indicative of the location of the object are received.Similarly, in order to obtain more information on the location of theobject, the exposure time must be increased. The methods and systemsdescribed herein may decouple the time and the space components usingthe sensor systems along with the methods for capture described herein.

Generally, coordinates X and Y described herein are coordinates on thefocal plane of the various sensors. Furthermore, time (T) is used inassociation with the X and Y coordinates to generally describe the X andY location of a light event, or event, at time T, wherein the lightevent comprises pixel illumination or a photon impact detection from aspace object. Furthermore, “light” may be electromagnetic radiation ofany wavelength. The techniques and detectors described herein work forlight reflected from a space object, or emitted by a space object,either from artificial or natural light sources. Depending on thedetectors, a single photon or light change event-based camera detectoras described herein, could be used from the Extreme Ultraviolet (or evenx-rays) to the long wave infrared. As such, a light wavelength band from100 angstroms to 10 microns or broader could be detected. For example,the techniques described herein can work for ultraviolet light from thesun reflected off any space object, ultraviolet light emitted fromthrusters on the space object, an artificial light (LED etc) on thespace object, natural visible light from the sun reflected off the spaceobject, lasers illuminating the space object, infrared radiation emittedby the heat of the space object, and/or the like.

In some embodiments, image photon counters (IPC) may be used. Exemplarymicrochannel plate (MCP) 302 delay line detector may be used. MCP 302delay line detector may asynchronously detect photon position in X and Yand store a corresponding time T associated with each photon impactevent while maintaining continuous spatial resolution. Similarly, singlephoton avalanche diode array (SPAD) 304 may asynchronously detectindividual photons in X and Y, and accurately measure a correspondingtime of impact. SPAD 304 may be limited in spatial resolution by pixelcount; however, each pixel is independent. Independent operation of eachpixel allows for simultaneous detection of photons across the focalplane. IPC sensors are discussed in more detail below.

In some embodiments, event-based sensor, which may be an event-basedcamera (EBC) 306, may be used. The exemplary EBC 306 illustrated is aneuromorphic camera. Asynchronous X, Y, P (polarity change), and timemay be detected. Spatial resolution may be defined by pixel count. Forevent-based sensors described herein, light changes above a minimumthreshold can be detected. This limits the data by only generating datawhen light illumination changes are detected rather than simultaneouslycollecting data from all pixels.

In some embodiments, a charged coupled device (CCD, not shown) or acomplementary metal oxide semiconductor (CMOS) 308 may be used. CCD andCMOS 308 are framing cameras that present the tradeoff between spatialaccuracy and timing accuracy described above. In some embodiments, ahigh frame rate in low light conditions must be used. Therefore, therecan be a high data volume penalty.

The sensors shown in FIG. 3 are exemplary only. Any sensors that collectinformation indicative of position, angular velocity, and angularacceleration may be used. Furthermore, various sensors may be used incombination. The sensors described above are further described inembodiments below for detecting angular velocity and angularacceleration.

IPC and EBC 306 may allow angular rate and angular acceleration to bemeasured. Furthermore, the data gathered by these sensors can betransformed without distortion into any fixed or co-moving coordinatesystem. Therefore, any measurements can be used between any relativelyfixed body or plurality of relatively fixed bodies and/or between aplurality of moving bodies.

Furthermore, frame sensor imaging, such as from charged couple devices(CCD), may collect photon data. However, the photon data may be subjectto error down to +/− one electron in low-light conditions. Therefore,even in low light conditions it is impossible to reliably collectinformation on individual photons impacting the frames. This results inphotons from the object being smeared out and diluted across the focalplane and mixed in with diffuse background light and starlight. Thetrack on the CCD output provides only angular position and time datafrom a plurality of measurements.

FIG. 4 depicts an exemplary comparison between IPC (e.g., MCP 302 andSPAD 304) and framing sensors (e.g., CCD, CMOS 308). IPC may collect X,Y, and time information by taking vibration measurements on membrane402. For example, photon 404 may impact the membrane causing a vibrationor sound wave across membrane 402 to membrane sensors 406 located atedges 408 of membrane 402. Membrane sensors 406 may detect the vibrationat different times. Knowing the propagation velocity of the vibrationthrough membrane 402 gives the precise location and time that photon 404impacted membrane 402. Therefore, the location and time data of thephoton impact are known. Time tagged photon list data can be projectedonto a plane 410 in a position-time three-dimensional space in which allthe photon events from the space object track 414 fall into a singlevirtual pixel 412 as shown in FIG. 4 . IPC allow rate tracking insoftware after the data is collected rather than during collection likethe CCD. As shown, the observation is a continuous data collection. Theobservations for the IPC are each photon that impacts membrane 402. Fromthe continuous data collection of the objects motion, higher orderderivatives of the angular motion are derivable through the entire spaceobject track 414. In the CCD, time delayed integration/orthogonaltransfer arrays the tracking rate must be known during observation ormechanical rate tracking. With IPC, movies with arbitrary timeresolutions may be created using the IPC data. This enhances movingobject detection as variable motion rates may be applied such that themotion of objects (e.g., stars) in the sky may be obscured while themotion of the object to be detected is enhanced. Furthermore, throughthese same methods, a plurality of objects with different and arbitraryvelocities may be detected in the same dataset.

Using the focal plane measurements (e.g., X and Y measurements) alongwith the precise time of the measurements, the angular velocity and theangular acceleration relative to the observer can be determined usingthe calculations presented in FIGS. 2A-2B. When the data is collectedand the angular velocity and the angular acceleration are determinedfrom the collected data, the angular velocity and the angularacceleration can be used to calculate the initial state vector using theabove-described modified Laplace then input into the orbit determinationalgorithms (e.g., extended Kalman Filer (EKF), Gooding method). Thisprovides a quick and efficient method of determining IOD when noise ispresent in the measurements.

Alternatively, in some embodiments, other sensors and techniques may beused to obtain the angular velocity and angular acceleration data of theobject. FIG. 5 depicts an exemplary scatterplot 500 generated from datacollected by EBC 306, for example. In some embodiments, EBC 306 may beused for detection of the objects. EBC 306 may record light levelchanges in pixels with high time resolution. EBC 306 may detect lightchanges above a threshold amount on each pixel and create a time stampfor each light change event. Accordingly, any change in the field ofview is recorded along with a highly accurate time that the changeoccurred. As each pixel only records data when a change is detected, theamount of data and processing is reduced and compared to a frame-basedcamera where all frames record at the same time. This allows each pixelto operate independently of the other pixels.

Allowing EBC 306 to collect data of a portion of the sky provides datathat may be visualized in exemplary scatterplot 500 in FIG. 5 . Thechanges in pixels are clearly shown as the various stars 502 are visibleas small lines caused by rotation of the earth and satellite 504 is adifferent length line caused by the satellites orbit. These lines aregenerated by the continuous detection light changes on each individualpixel and associated time data. Therefore, the object angular rate andangular acceleration may be directly determined from EBC 306 obtaineddata.

FIG. 6 depicts an embodiment of detecting intensity variations of theobject as the object moves through the orbit. In the arrangement shown,the light from the object is magnified by telescope 602 then transmittedthrough a system of lenses 604 and Ronchi filter 606. Ronchi filter 606may comprise varying slit widths to match the tangent plane angularmotion. As such, as the object moves across the focal plane, the lightintensity from the object passing through Ronchi filter 606 isindicative of the angular velocity of the object.

Ronchi filter 606 must have a variable slit pattern to match the angulardisplacements of the optical rays 608. Because of the tangent planeprojection of the image on the focal plane, the spacing and width ofRonchi filter 606 bars should increase toward the edge of the field ofview. This provides a openings of Ronchi filter 606 representing aconstant angular displacement. Therefore, the angular velocity isdirectly measured without compensation for the ray displacement anglesand is completely decoupled from the fixed angle measurements.

As the object moves across the focal plane, a low spatial resolutionhigh time resolution sensor 610 may be used to accurately detect thetime of the light event arrival and the light intensity of the light anddark bars as they change on the screen as shown at process 612. Thefrequency of change in the intensity 614 may be measured by spectralanalysis. The intensity frequency may be indicative of the angularvelocity and the change in intensity may be indicative of the angularacceleration. As such, the angular velocity and the angular accelerationmay be determined by the spectral analysis. The angular velocity and theangular acceleration uncertainty and covariance are decoupled from theabsolute angle measurements.

FIG. 7 depicts an embodiment utilizing color filters to encode x and ymotion and brightness. In some embodiments, three filters 702 may beused to block frequencies of light such that the motion encoding foreach X and Y component is shown. In an exemplary embodiment, magentafilter 704 blocks green light allowing red and blue, yellow filter 706blocks blue light allowing green and red, and a cyan filter (not shown)blocks red light passing green and blue. In some embodiments, thecombination of the magenta and yellow Ronchi filters encodes X and Ymotion across gratings on a single-color focal plane. A blue-green-redcamera may be used to detect the different color outputs. At block 708,blue represents the component of X motion, green represents thecomponent of Y motion, and red represents the intensity changes overtime. This allows the angular motion to brightness intensity encoding tobe performed with one telescope instead of three.

In some embodiments, additional algorithms (e.g., machine learningalgorithms) may be applied to improve object detection and fidelity ofresults. For example, a convolution neural network (CNN) may be trainedto characterize orbiting objects against the sky. Based on the time ofexposure and the length of track, the type of orbit and possibly thetype of satellite may be characterized by the CNN. Any artificialintelligence (AI) algorithm or statistical algorithm may be used tofurther enhance object detection.

FIG. 8 depicts an exemplary method of determining an initial orbit,generally referenced by the numeral 800. At step 802, the sensorreceives light from the sky including the orbiting object as describedin embodiments above. The sensor may be any IPC, EBC, telescope sensorarrangement as describe above, and/or any other arrangement andcombination that may be capable of detecting angular velocity andangular acceleration.

At step 804, the sensor processes the received light as described inembodiments above. Depending on the sensor that is being used theprocessing may be different. Here, the sensors may detect light eventsincluding frequency variations, intensity variations, illuminationchanges, and individual photons coupled with accurate time measurementsto accurately determine angular position, angular velocity and angularacceleration of the detected orbital objects.

At step 806, the angular velocity and the angular acceleration may bedetermined from the received data. The received data collected by thesensors may be indicative of the angular velocity and the angularacceleration, but the data may be time and position data of photons,frequency change rates, intensity change rates, and the like asdescribed above. A curve may be fit to the data indicative of theangular velocity and the angular acceleration. The angular velocity andangular acceleration may be determined from the curve fit. In someembodiments, the right ascension and the declination may be determined.

At step 808, the angular velocity and the angular acceleration may beinput into the above-described algorithms (e.g., modified Laplace) todetermine range and range rate between the target object and theobserver. Furthermore, the initial state vector may be determined usingthe range and the range rate. The IOD is more quickly determined overcurrent methods by inputting the angular velocity and the angularacceleration into the range and range rate equations described in FIGS.2A and 2B. The initial state vector may then be used in the EKF or Batchalgorithm at step 810. When noise is present in the real-worldmeasurements, the above-described decoupled systems produce aconvergence on the accurate orbit by a magnitude of at least one overcurrent methods.

FIG. 9 depicts a more complete process of utilizing the sensors andalgorithms for calibrating sensors, employing the IOD process describedabove, and using the IOD output for orbit determination. In someembodiments, exquisite calibration may be necessary or useful inproviding an efficient IOD. In the calibration process may comprise alist-based process for generating a dynamic model of the sensor field ofview. In some embodiments, the data may be conditioned for calculationin a pre-processing phase. In some embodiments, the sensor data may notbe correct for calculation of astronomical coordinates. For example, theX and Y data on the focal plane, angular rate, angular acceleration, andso forth, may be scaled based on the sensor or telescopic parameters. Arescaling of the data may be used to generate a pseudo—X′, Y′ to producea round and centered focal plane image. The data may be conditioned anddivided up into one second time sections and saved as a flexible imagetransport system (FITS) image. The plurality of images may then beanalyzed, solving for the astrometric coordinate system. In someembodiments, this process may be performed by a third party such as, forexample, astronomy.net. The process is described in detail below.

Step 902 may begin the pre-processing routine to gather star data andcreate a corrected star data space. At step 902, the stars across thesky may be detected and the velocity moving across focal plane vx, vy,as described above, may be determined in pixel coordinates. In someembodiments, a best instrument arrangement for capturing the object maybe determined. The best setting may be sensor specific. For example, thebest setting may be a telescope orientation that is fixed. Accordingly,the sky will affectively move past the field of view of the telescopeand the sensor. As such, the stars will trace a path across the field ofview as will the objects in orbit. In some embodiments, it may be betterto have the telescope move or rotate with the stars, so the stars areapparently motionless in the field of view and the only perceivedmovement are the orbiting objects. Furthermore, the telescope andsensors are arranged to detect the desired portion of the sky comprisingthe orbiting objects with known locations that may be compared later.For the exemplary process described herein, the telescope is stationaryand corrections are made for the relative motion of the space objects.

A set of data may be received by the sensor. The sensor may be operatedto receive the light from the sky to detect all luminous objects in thedesired locations. The light is collected by the sensor. The sensor maybe any of the above-described sensors and the light may be collected byany of the above-described systems and methods associated with thesensors.

At step 904, a virtual image (2-D histogram) of events may be cast inthe star motion corrected space for sidereal tracking including therotation of the earth and aberrations due to the relative velocity ofearth. The star motion corrected space may be defined by X′=x+vx*t;Y′=y+vy*t.

At step 906, astrometric processing pass one begins. At step 906, pointsources are selected in the virtual image and tabulated foridentification and classification. The list-based approach describedherein provides tracking data such that the data is not bent. Thetracking data is added to the list as it becomes more and more refinedas described in more detail below.

At step 908, the point source solutions are realized. The list of pointsource data may be submitted to an online database such as for example,astronomy.net, for processing. The astrometric solutions are provided inX′, Y′ space.

At step 910, all events may be transformed into to RA and DEC datacoordinates using the solutions in X′, Y′ space. All events are thenlabeled with the RA and DEC data.

At step 912, a search in the GAZA catalogue may be conducted forrelevant stars in the virtual image area of the sky. At step 914, GAZAstars are assigned to the source objects in the virtual image. At step916, the GAZA catalogue coordinates are corrected for aberration databased on the Earth's velocity. At step 918, all events associate withthe image sources are labeled with the corrected GAZA coordinates.

Astrometric processing pass two begins at step 920. At step 920, aseventeen-parameter time dependent dynamic model of the astrometrictangent plane is then fit to map each event x, y, time to RA and DEC.The mapping provides an x, y, and time stamp to each event and atransformation to RA and DEC. Furthermore, an optical barrel correctionmay be performed to correct for spherical mapping. A simultaneous fit isperformed for space objects. This process defines how the field of viewis moving across the sky relative to the sensor mount rotating with theEarth. The pixels coordinates may be fixed to right ascension and time.The list process continues adding more and more refined data to the listrather than bending the data to fit the mapping. This dynamic model fitis used to confirm the data and label the data as pass two RA and DECposition on the sky. This allows for quick transformation between X, Y,and time and RA, DEC, and time. At step 922, all events are labeled withthe results of the pass 2 tangent plane motion fit.

Pass three astrometric processing begins at step 924. At step 924, theRA and DEC residuals for all events may be associated with the GAZAcatalog stars may be plotted. At step 926, a spline fit may be generatedfor the for RA and DEC corrections in time. This may be used to correctthe data for unmodeled inputs such as, for example, telescope mountjitter. At step 928, all events may be corrected with pass threecorrections. These corrections provide real event location dataindicative of the locations on the sky. At this point all events can belabeled as stars or the object (e.g., satellite, meteor, missile, or thelike).

At step 930, space object event identification as described inembodiments above may begin. At step 930, the x and y pixel velocity ofthe object may be determined from the sensor data. Once the x and ypixel velocity are identified, the virtual rate track image may begenerated at step 932 identifying the source. The virtual rate trackimage may be indicative of the x and y velocity of the source over timeand the events associated with the source may be tracked over time atstep 934. The source of the light impacts may be compared to identifythe object based on the rate track as described in embodiments above.

The above-described IOD process may begin at step 936. At step 936, RAand DEC of space objects may be fit to polynomials and splines. Themiddle time of the space object event set may be determined and theangle, angular rate, and angular acceleration at that time may bedetermined at step 938.

The IOD algorithm may be applied at step 940. At step 940, the range andrange rate may be computed using the measured angular velocity andangular acceleration. Furthermore, the other orbital elements may becomputed from the IOD. In some embodiments, the other orbital elementsmay be the orbital elements necessary to describe the orbit as shown inFIGS. 2A and 2B and described in detail above.

At step 942 an extension to the IOD may be added, referenced herein asIOD+. At step 942, an exemplary Levenberg-Marquardt algorithm may beused in conjunction with the IOD to fit an orbit ephemeris to the eventdata. The orbit ephemeris may be used as a starting seed solution fororbit determination. This provides a better starting seed at a quickertime than current standard methods of orbit determination.

At step 944, typical orbit determination techniques may be used toestimate an orbit of the space object using the starting seed solution.

In FIG. 10 , an exemplary hardware platform 1000 for certain embodimentsof the invention is depicted. Computer 1002 can be a desktop computer, alaptop computer, a server computer, a mobile device such as a smartphoneor tablet, or any other form factor of general- or special-purposecomputing device. Depicted with computer 1002 are several components,for illustrative purposes. In some embodiments, certain components maybe arranged differently or absent. Additional components may also bepresent. Included in computer 1002 is system bus 1004, whereby othercomponents of computer 1002 can communicate with each other. In certainembodiments, there may be multiple busses or components may communicatewith each other directly. Connected to system bus 1004 is centralprocessing unit (CPU) 1006. Also attached to system bus 1004 are one ormore random-access memory (RAM) modules 1008. Also attached to systembus 1004 is graphics card 1010. In some embodiments, graphics card 1010may not be a physically separate card, but rather may be integrated intothe motherboard or the CPU 1006. In some embodiments, graphics card 1010has a separate graphics-processing unit (GPU) 1012, which can be usedfor graphics processing or for general purpose computing (GPGPU). Alsoon graphics card 1010 is GPU memory 1014. Connected (directly orindirectly) to graphics card 1010 is display 1016 for user interaction.In some embodiments, no display is present, while in others it isintegrated into computer 1002. Similarly, peripherals such as keyboard1018 and mouse 1020 are connected to system bus 1004. Like display 1016,these peripherals may be integrated into computer 1002 or absent. Alsoconnected to system bus 1004 is local storage 1022, which may be anyform of computer-readable media and may be internally installed incomputer 1002 or externally and removably attached.

Computer-readable media include both volatile and nonvolatile media,removable and nonremovable media, and contemplate media readable by adatabase. For example, computer-readable media include (but are notlimited to) RAM, ROM, EEPROM, flash memory or other memory technology,CD-ROM, digital versatile discs (DVD), holographic media or otheroptical disc storage, magnetic cassettes, magnetic tape, magnetic diskstorage, and other magnetic storage devices. These technologies canstore data temporarily or permanently. However, unless explicitlyspecified otherwise, the term “computer-readable media” should not beconstrued to include physical, but transitory, forms of signaltransmission such as radio broadcasts, electrical signals through awire. Examples of stored information include computer-usableinstructions, data structures, program modules, and other datarepresentations.

Finally, network interface card (NIC) 1024 is also attached to systembus 1004 and allows computer 1002 to communicate over a network such asnetwork 1026. NIC 1024 can be any form of network interface known in theart, such as Ethernet, ATM, fiber, Bluetooth, or Wi-Fi (i.e., the IEEE802.11 family of standards). NIC 1024 connects computer 1002 to localnetwork 1026, which may also include one or more other computers, suchas computer 1028, and network storage, such as data store 1030.Generally, a data store such as data store 1030 may be any repositoryfrom which information can be stored and retrieved as needed. Examplesof data stores include relational or object-oriented databases,spreadsheets, file systems, flat files, directory services such as LDAPand Active Directory, or email storage systems. A data store may beaccessible via a complex API (such as, for example, Structured QueryLanguage), a simple API providing only read, write and seek operations,or any level of complexity in between. Some data stores may additionallyprovide management functions for data sets stored therein such as backupor versioning. Data stores can be local to a single computer such ascomputer 1028, accessible on a local network such as local network 1026,or remotely accessible over Internet 1032. Local network 1026 is in turnconnected to Internet 1032, which connects many networks such as localnetwork 1026, remote network 1034 or directly attached computers such ascomputer 1036. In some embodiments, computer 1002 can itself be directlyconnected to Internet 1032.

In some aspects, the techniques described herein relate to a method ofdetermining an orbit of an object, the method including: obtaining, by asensor, data indicative of a plurality of objects in orbits, wherein thedata is indicative of angular velocity and angular acceleration of theobject of the plurality of objects, identifying the object from thedata, measuring the angular velocity and the angular acceleration fromthe data, determining an initial orbit of the object using the angularvelocity and the angular acceleration, and estimating orbital elementsfrom the initial orbit of the object.

In some aspects, the method further includes detecting locations oflight impacts on pixels of the sensor, wherein the locations of thelight impacts are indicative of relative locations of the plurality ofobjects in a field of view; generating a virtual rate track image of thelocations of the light impacts over time; and identifying the object andan associated angle, the angular velocity, and the angular accelerationfrom the virtual rate track image.

In some aspects, the techniques described herein relate to the method,wherein the sensor is an event-based camera, and wherein the data isindicative of a change in luminosity on each pixel of the event-basedcamera.

In some aspects, the techniques described herein relate to the method,wherein the sensor is an image photon counter, and the data isindicative of a time and a location of photon impacts on the sensor.

In some aspects, the techniques described herein relate to the method,wherein the sensor is a color sensor and incoming light is filtered tobe indicative of components of the angular velocity and the angularacceleration.

In some aspects, the techniques described herein relate to the method,wherein the sensor is an earth-based sensor or a space-based sensor.

In some aspects, the techniques described herein relate to one or morenon-transitory computer-readable media storing computer-executableinstructions that, when executed by at least one processor, performs amethod of determining an orbit of an object. The method includingobtaining, by a sensor, data indicative of a plurality of objects inorbits, wherein the data is indicative of angular velocity and angularacceleration of the object of the plurality of objects, identifying theobject from the data by: generating a virtual rate track image includingvirtual rate tracks of each object of the plurality of objects in thevirtual rate track image, mapping the virtual rate tracks to locationsin a field of view, identifying the plurality of objects based on thelocations in the field of view, and identifying the object from theplurality of objects based on a comparison of the virtual rate tracks;and determining the angular velocity and the angular acceleration of theobject from the virtual rate tracks, determining an initial orbit of theobject using the angular velocity and the angular acceleration, andestimating orbital elements from the initial orbit of the object.

In some aspects, the techniques described herein relate to the media,wherein the method further includes determining an initial state vectorincluding range and range rate of the object determined from the angularvelocity and the angular acceleration.

In some aspects, the techniques described herein relate to the media,further including: determining the initial orbit of the object byfitting an orbit ephemeris to the data; and determining the orbit of theobject using the initial orbit as a starting seed solution.

In some aspects, the techniques described herein relate to the media,wherein the orbit ephemeris is fitted to the data using aLevenberg-Marquardt algorithm.

In some aspects, the techniques described herein relate to the media,wherein the sensor is one of a microchannel plate delay line detector,event-based camera, single photon avalanche diode array, or acomplementary metal oxide semiconductor.

In some aspects, the techniques described herein relate to the media,wherein the angular velocity, the angular acceleration, and anassociated angle are determined at a middle time of the data.

In some aspects, the techniques described herein relate to the media,wherein the sensor is a positioned in space.

In some aspects, the techniques described herein relate to one or morenon-transitory computer-readable media storing computer-executableinstructions that, when executed by at least one processor, performs amethod of determining an orbit of an object. The method includesobtaining, by a sensor, data indicative of a plurality of objects inorbits, wherein the data is indicative of angular velocity and angularacceleration of the object of the plurality of objects, identifying theobject from the data; measuring the angular velocity and the angularacceleration from the data, determining an initial orbit of the objectby fitting an orbit ephemeris to the data; and determining the orbit ofthe object using the initial orbit as a starting seed solution for anorbit determination algorithm.

In some aspects, the techniques described herein relate to the media,wherein the sensor is one of a microchannel plate delay line detector,event-based camera, single photon avalanche diode array, or acomplementary metal oxide semiconductor.

In some aspects, the techniques described herein relate to the media,wherein the method further includes determining an x and y pixelvelocity of the object on the sensor.

In some aspects, the techniques described herein relate to the media,wherein the identifying the object from the data includes: generating avirtual rate track image from the x and y pixel velocity, the virtualrate track image including virtual rate tracks of each object of theplurality of objects in the virtual rate track image, mapping thevirtual rate tracks to locations in a field of view; and comparing eachrate track of the virtual rate tracks to determine a rate trackcorresponding to the object.

In some aspects, the techniques described herein relate to the media,wherein the angular velocity, the angular acceleration, and anassociated angle are determined at a middle time of the data.

In some aspects, the techniques described herein relate to the media,wherein the orbit ephemeris is fitted to the data using aLevenberg-Marquardt algorithm.

Although the invention has been described with reference to theembodiments illustrated in the attached drawing figures, it is notedthat equivalents may be employed, and substitutions made herein withoutdeparting from the scope of the invention as recited in the claims.

Having thus described various embodiments of the disclosure, what isclaimed as new and desired to be protected by Letters Patent includesthe following:

1. A method of determining an orbit of an object, the method comprising:obtaining, by a sensor, data indicative of a plurality of objects inorbits; wherein the data is indicative of angular velocity and angularacceleration of the object of the plurality of objects; detectinglocations of light events on pixels of the sensor, wherein the locationsof the light events are indicative of relative locations of theplurality of objects in a field of view; tracking the plurality ofobjects over time; identifying the object from the tracking; determiningthe angular velocity and the angular acceleration of the object from thetracking; determining an initial orbit of the object using the angularvelocity and the angular acceleration; and estimating orbital elementsfrom the initial orbit of the object.
 2. The method of claim 1, furthercomprising: generating a virtual rate track image of the locations ofthe light events over time; and identifying the object and an associatedangle, the angular velocity, and the angular acceleration from thevirtual rate track image.
 3. The method of claim 2, wherein the sensoris an event-based camera, and wherein the data is indicative of a changein luminosity on each pixel of the event-based camera.
 4. The method ofclaim 2, wherein the sensor is an image photon counter, and the data isindicative of the times and the locations of photon impacts on thesensor.
 5. The method of claim 1, wherein the sensor is a color sensorand incoming light is filtered to be indicative of components of theangular velocity and the angular acceleration.
 6. The method of claim 1,wherein the sensor is an earth-based sensor.
 7. The method of claim 1,wherein the sensor is a space-based sensor.
 8. One or morenon-transitory computer-readable media storing computer-executableinstructions that, when executed by at least one processor, performs amethod of determining an orbit of an object, the method comprising:obtaining, by a sensor, data indicative of a plurality of objects inorbits; wherein the data is indicative of angular velocity and angularacceleration of the object of the plurality of objects; identifying theobject from the data by: generating a virtual rate track imagecomprising virtual rate tracks of each object of the plurality ofobjects in the virtual rate track image, mapping the virtual rate tracksto locations in a field of view; identifying the plurality of objectsbased on the locations in the field of view; and identifying the objectfrom the plurality of objects based on a comparison of the virtual ratetracks; and determining the angular velocity and the angularacceleration of the object from the virtual rate tracks; determining aninitial orbit of the object using the angular velocity and the angularacceleration; and estimating orbital elements from the initial orbit ofthe object.
 9. The media of claim 8, wherein the method furthercomprises determining an initial state vector comprising range and rangerate of the object determined from the angular velocity and the angularacceleration.
 10. The media of claim 8, further comprising: determiningthe initial orbit of the object by fitting an orbit ephemeris to thedata; and determining the orbit of the object using the initial orbit asa starting seed solution.
 11. The media of claim 10, wherein the orbitephemeris is fitted to the data using a Levenberg-Marquardt algorithm.12. The media of claim 8, wherein the sensor is one of a microchannelplate delay line detector, event-based camera, single photon avalanchediode array, or a complementary metal oxide semiconductor.
 13. The mediaof claim 8, wherein the angular velocity, the angular acceleration, andan associated angle are determined at a middle time of the data.
 14. Themedia of claim 8, wherein the sensor is a positioned in space.
 15. Oneor more non-transitory computer-readable media storingcomputer-executable instructions that, when executed by at least oneprocessor, performs a method of determining an orbit of an object, themethod comprising: obtaining, by a sensor, data indicative of aplurality of objects in orbits; wherein the data is indicative ofangular velocity and angular acceleration of the object of the pluralityof objects; identifying the object from the data; measuring the angularvelocity and the angular acceleration from the data; determining aninitial orbit of the object by fitting an orbit ephemeris to the data;and determining the orbit of the object using the initial orbit as astarting seed solution for an orbit determination algorithm.
 16. Themedia of claim 15, wherein the sensor is one of a microchannel platedelay line detector, event-based camera, single photon avalanche diodearray, or a complementary metal oxide semiconductor.
 17. The media ofclaim 16, wherein the method further comprises determining an x and ypixel velocity of the object on the sensor.
 18. The media of claim 17,wherein the identifying the object from the data comprises: generating avirtual rate track image from the x and y pixel velocity, the virtualrate track image comprising virtual rate tracks of each object of theplurality of objects in the virtual rate track image, mapping thevirtual rate tracks to locations in a field of view; and comparing eachrate track of the virtual rate tracks to determine a rate trackcorresponding to the object.
 19. The media of claim 18, wherein theangular velocity, the angular acceleration, and an associated angle aredetermined at a middle time of the data.
 20. The media of claim 19,wherein the orbit ephemeris is fitted to the data using aLevenberg-Marquardt algorithm.